Bayesian GLM Part3
Murray Logan
08 October, 2022
Preparations
Load the necessary libraries
library(tidyverse) #for data wrangling etc
library(rstanarm) #for fitting models in STAN
library(cmdstanr) #for cmdstan
library(brms) #for fitting models in STAN
library(standist) #for exploring distributions
library(coda) #for diagnostics
library(bayesplot) #for diagnostics
library(ggmcmc) #for MCMC diagnostics
library(DHARMa) #for residual diagnostics
library(rstan) #for interfacing with STAN
library(emmeans) #for marginal means etc
library(broom) #for tidying outputs
library(tidybayes) #for more tidying outputs
library(HDInterval) #for HPD intervals
library(ggeffects) #for partial plots
library(broom.mixed)#for summarising models
library(posterior) #for posterior draws
library(ggeffects) #for partial effects plots
library(patchwork) #for multi-panel figures
library(bayestestR) #for ROPE
library(see) #for some plots
theme_set(theme_grey()) #put the default ggplot theme back
source('helperFunctions.R')Scenario
Here is a modified example from Peake and Quinn (1993). Peake and Quinn (1993) investigated the relationship between the number of individuals of invertebrates living in amongst clumps of mussels on a rocky intertidal shore and the area of those mussel clumps.
Format of peakquinn.csv data files
| AREA | INDIV |
|---|---|
| 516.00 | 18 |
| 469.06 | 60 |
| 462.25 | 57 |
| 938.60 | 100 |
| 1357.15 | 48 |
| ... | ... |
| AREA | Area of mussel clump mm2 - Predictor variable |
| INDIV | Number of individuals found within clump - Response variable |
The aim of the analysis is to investigate the relationship between mussel clump area and the number of non-mussel invertebrate individuals supported in the mussel clump.
Read in the data
peake <- read_csv('../public/data/peakquinn.csv', trim_ws=TRUE)
glimpse(peake)## Rows: 25
## Columns: 2
## $ AREA <dbl> 516.00, 469.06, 462.25, 938.60, 1357.15, 1773.66, 1686.01, 1786.…
## $ INDIV <dbl> 18, 60, 57, 100, 48, 118, 148, 214, 225, 283, 380, 278, 338, 274…Exploratory data analysis
When exploring these data as part of a frequentist analysis, exploratory data analysis revealed that the both the response (counds of individuals) and predictor (mussel clump area) were skewed and the relationship between raw counds and mussel clump area was not linear. Furthermore, there was strong evidence of a relationship between mean and variance. Normalising both reponse and predictor addressed these issues. However, rather than log transform the response, it was considered more appropriate to model against a distribution that used a logarithmic link function.
The individual observations here (\(y_i\)) are the observed number of (non mussel individuals found in mussel clump \(i\). As a count, these might be expected to follow a Poisson (or perhaps negative binomial) distribution. In the case of a negative binomial, the observed count for any given mussel clump area are expected to be drawn from a negative binomial distribution with a mean of \(\lambda_i\). All the negative binomial distributions are expected to share the same degree of dispersion (\(\theta\)) - that is, the degree to which the inhabitants of mussell clumps aggregate together (or any other reason for overdispersion) is independent of mussel clump area and can be estimated as a constant across all populations.
The natural log of the expected values (\(\lambda_i\)) is modelled against a linear predictor that includes an intercept (\(\beta_0\)) and a slope (\(\beta_1\)) associated with the natural log of mussel area.
The priors on \(\beta_0\) and \(\beta_1\) should be on the natura log scale (since this will be the scale of the parameters). As starting points, we will consider the following priors:
- \(\beta_0\): Normal prior centered at 0 with a variance of 5
- \(\beta_1\): Normal prior centered at 0 with a variance of 2
- \(\theta\): Exponential prior with rate 1
Model formula: \[ \begin{align} y_i &\sim{} \mathcal{NB}(\lambda_i, \theta)\\ ln(\lambda_i) &= \beta_0 + \beta_1 ln(x_i)\\ \beta_0 & \sim\mathcal{N}(6,1.5)\\ \beta_1 & \sim\mathcal{N}(0,1)\\ \theta &\sim{} \mathcal{Exp}(1)\\ OR\\ \theta &\sim{} \mathcal{\Gamma}(2,1)\\ \end{align} \]
Fit the model
frequentist
summary(glm(INDIV ~ log(AREA), data=peake, family=poisson()))##
## Call:
## glm(formula = INDIV ~ log(AREA), family = poisson(), data = peake)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -16.717 -3.614 -1.679 1.808 21.744
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.013528 0.091147 0.148 0.882
## log(AREA) 0.691628 0.009866 70.100 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 7494.5 on 24 degrees of freedom
## Residual deviance: 1547.8 on 23 degrees of freedom
## AIC: 1738.9
##
## Number of Fisher Scoring iterations: 4summary(MASS::glm.nb(INDIV ~ log(AREA), data=peake))##
## Call:
## MASS::glm.nb(formula = INDIV ~ log(AREA), data = peake, init.theta = 7.369263003,
## link = log)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -2.29133 -0.59599 -0.06927 0.48929 1.64734
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.16452 0.53387 -2.181 0.0292 *
## log(AREA) 0.82469 0.06295 13.102 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for Negative Binomial(7.3693) family taken to be 1)
##
## Null deviance: 161.076 on 24 degrees of freedom
## Residual deviance: 25.941 on 23 degrees of freedom
## AIC: 312.16
##
## Number of Fisher Scoring iterations: 1
##
##
## Theta: 7.37
## Std. Err.: 2.16
##
## 2 x log-likelihood: -306.16rstanarm
Using default priors
In rstanarm, the default priors are designed to be
weakly informative. They are chosen to provide moderate regularlization
(to help prevent overfitting) and help stabalise the computations.
peake.rstanarm = stan_glm(INDIV ~ log(AREA), data=peake,
family=poisson(),
iter = 5000, warmup = 1000,
chains = 3, thin = 5, refresh = 0)prior_summary(peake.rstanarm)## Priors for model 'peake.rstanarm'
## ------
## Intercept (after predictors centered)
## ~ normal(location = 0, scale = 2.5)
##
## Coefficients
## Specified prior:
## ~ normal(location = 0, scale = 2.5)
## Adjusted prior:
## ~ normal(location = 0, scale = 2)
## ------
## See help('prior_summary.stanreg') for more detailsThis tells us:
for the intercept, it is using a normal prior with a mean of 0 and a standard deviation of 2.5. The are the defaults for all non-Gaussian intercepts.
for the coefficients (in this case, just the slope), the default prior is a normal prior centered around 0 with a standard deviation of 2.5. For Poisson models, this is then adjusted for the scale of the data by dividing the 2.5 by the standard deviation of the (in this case log) predictor (then rounded).
2.5/sd(log(peake$AREA))## [1] 2.025039- there is no auxillary parameter and thus there is no auxillary prior
Assessing priors
One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging preditions would ensure that the priors are unlikely to influence the actual preditions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of paramter space that are not supported by the actual data, the sampler can become unstable and have difficulty.
We can draw from the prior predictive distribution instead of
conditioning on the response, by updating the model and indicating
prior_PD=TRUE. After refittin the model in this way, we can
plot the predictions to gain insights into the range of predictions
supported by the priors alone.
peake.rstanarm1 <- update(peake.rstanarm, prior_PD=TRUE)ggemmeans(peake.rstanarm1, ~AREA) %>% plot(add.data=TRUE)
ggemmeans(peake.rstanarm1, ~AREA) %>% plot(jitter = FALSE, add.data = TRUE, log.y = TRUE)
ggemmeans(peake.rstanarm1, ~AREA) %>% plot(jitter = FALSE, add.data = TRUE) + scale_y_log10()
Conclusions:
- we see that although the range of predictions are very wide and the slope could range from strongly negative to strongly positive, the choice to have the intercept prior have a mean of 0 results in most of the prior-only posterior draws being dramatically lower than the observed values - indeed some observed values are above the range of the prior-posterior predictions.
Defining priors
The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]
When defining our own priors, we typically do not want them to be scaled.
If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucket out of thin air):
- \(\beta_0\): normal centred at 6
with a standard deviation of 6
- mean of 6: because (
mean(log(peake$INDIV))) - sd of 2.8: because (
2.5 * sd(log(peake$INDIV)))
- mean of 6: because (
- \(\beta_1\): normal centred at 0
with a standard deviation of 1
- sd of 2.3: because
(
sd(log(peake$INDIV))/sd(log(peake$AREA)))
- sd of 2.3: because
(
I will also overlay the raw data for comparison.
peake.rstanarm2 <- stan_glm(INDIV ~ log(AREA), data=peake,
family=poisson(),
prior_intercept = normal(6, 2.8, autoscale=FALSE),
prior = normal(0, 2.3, autoscale=FALSE),
prior_PD=TRUE,
iter = 5000, warmup = 1000,
chains = 3, thin = 5, refresh = 0
)ggemmeans(peake.rstanarm2, ~AREA) %>%
plot(add.data=TRUE) +
scale_y_log10()
Now lets refit, conditioning on the data.
peake.rstanarm3 <- update(peake.rstanarm2, prior_PD=FALSE) Plotting prior and posterior
posterior_vs_prior(peake.rstanarm3, color_by='vs', group_by=TRUE,
facet_args=list(scales='free_y'))
Conclusions:
- in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.
ggemmeans(peake.rstanarm3, ~AREA) %>% plot(add.data=TRUE)
brms
Using default priors
In brms, the default priors are designed to be weakly
informative. They are chosen to provide moderate regularlization (to
help prevent overfitting) and help stabalise the computations.
Unlike rstanarm, brms models must be
compiled before they start sampling. For most models, the compilation of
the stan code takes around 45 seconds.
peake.brm <- brm(bf(INDIV ~ log(AREA), family=poisson()),
data=peake,
iter = 5000, warmup = 1000,
chains = 3, cores = 3,
thin = 5, refresh = 0,
backend = "cmdstan")## Running MCMC with 3 parallel chains...
##
## Chain 1 finished in 0.1 seconds.
## Chain 2 finished in 0.1 seconds.
## Chain 3 finished in 0.1 seconds.
##
## All 3 chains finished successfully.
## Mean chain execution time: 0.1 seconds.
## Total execution time: 0.3 seconds.prior_summary(peake.brm)## prior class coef group resp dpar nlpar lb ub source
## (flat) b default
## (flat) b logAREA (vectorized)
## student_t(3, 5.8, 2.5) Intercept defaultThis tells us:
- for the intercept, it is using a student t (flatter normal) prior with a mean of 5.8 and a standard deviation of 2.5. The mean of this prior is based on the median of the log-transformed response and the standard deviation of 2.5 is the default minimum.
## [1] 5.823046mad(log(peake$INDIV))## [1] 1.025465for the beta coefficients (in this case, just the slope), the default prior is a inproper flat prior. A flat prior essentially means that any value between negative infinity and positive infinity are equally likely. Whilst this might seem reckless, in practice, it seems to work reasonably well for non-intercept beta parameters.
there is no sigma parameter in a binomial model and therefore there are no additional priors.
Assessing priors
One way to assess the priors is to have the MCMC sampler sample purely from the prior predictive distribution without conditioning on the observed data. Doing so provides a glimpse at the range of predictions possible under the priors. On the one hand, wide ranging preditions would ensure that the priors are unlikely to influence the actual preditions once they are conditioned on the data. On the other hand, if they are too wide, the sampler is being permitted to traverse into regions of parameter space that are not logically possible in the context of the actual underlying ecological context. Not only could this mean that illogical parameter estimates are possible, when the sampler is traversing regions of paramter space that are not supported by the actual data, the sampler can become unstable and have difficulty.
In brms, we can inform the sampler to draw from the
prior predictive distribution instead of conditioning on the response,
by running the model with the sample_prior='only' argument.
Unfortunately, this cannot be applied when there are flat priors (since
the posteriors will necessarily extend to negative and positive
infinity). Therefore, in order to use this useful routine, we need to
make sure that we have defined a proper prior for all parameters.
We will adopt a similar approach to that of rstanarm -
that is 2.5 * sd(log(peake$INDIV))/sd(log(peake$AREA))
(approx. 2.3).
peake.brm1 = brm(bf(INDIV ~ log(AREA), family=poisson()),
data=peake,
prior=c(
prior(normal(0, 2.3), class='b')),
sample_prior = 'only',
iter = 5000, warmup = 1000,
chains = 3, cores = 3,
thin = 5, refresh = 0,
backend = "cmdstan")## Running MCMC with 3 parallel chains...
##
## Chain 1 finished in 0.0 seconds.
## Chain 2 finished in 0.0 seconds.
## Chain 3 finished in 0.0 seconds.
##
## All 3 chains finished successfully.
## Mean chain execution time: 0.0 seconds.
## Total execution time: 0.2 seconds.ggemmeans(peake.brm1, ~AREA) %>% plot(add.data=TRUE)
ggemmeans(peake.brm1, ~AREA) %>% plot(add.data=TRUE, log.y = TRUE)
ggemmeans(peake.brm1, ~AREA) %>% plot(add.data=TRUE) + scale_y_log10()
conditional_effects(peake.brm1) %>% plot(points=TRUE) %>% `[[`(1) + scale_y_log10()
plot(conditional_effects(peake.brm1), points=TRUE, plot = FALSE)[[1]] + scale_y_log10()
Conclusions:
- we see that the range of predictions is faily wide and the slope could range from strongly negative to strongly positive.
Defining priors
The following link provides some guidance about defining priors. [https://github.com/stan-dev/stan/wiki/Prior-Choice-Recommendations]
When defining our own priors, we typically do not want them to be scaled.
If we wanted to define our own priors that were less vague, yet still not likely to bias the outcomes, we could try the following priors (mainly plucket out of thin air):
- \(\beta_0\): normal centred at 6
with a standard deviation of 6
- mean of 6: because (
mean(log(peake$INDIV))) - sd of 1.5: because (
sd(log(peake$INDIV)))
- mean of 6: because (
- \(\beta_1\): normal centred at 0
with a standard deviation of 1
- sd of 1: because
(
sd(log(peake$INDIV))/sd(log(peake$AREA)))
- sd of 1: because
(
I will also overlay the raw data for comparison.
peake.brm2 <- brm(bf(INDIV ~ log(AREA), family=poisson()),
data=peake,
prior=c(
prior(normal(6, 1.5), class='Intercept'),
prior(normal(0, 1), class='b')
),
sample_prior = 'only',
iter = 5000, warmup = 1000,
chains = 3, cores = 3,
thin = 5, refresh = 0,
backend = "cmdstan")## Running MCMC with 3 parallel chains...
##
## Chain 1 finished in 0.0 seconds.
## Chain 2 finished in 0.0 seconds.
## Chain 3 finished in 0.0 seconds.
##
## All 3 chains finished successfully.
## Mean chain execution time: 0.0 seconds.
## Total execution time: 0.2 seconds.ggemmeans(peake.brm2, ~AREA) %>%
plot(add.data=TRUE) +
scale_y_log10()
peake.brm3 <- update(peake.brm2, sample_prior=TRUE, refresh=0) ## Running MCMC with 3 sequential chains...
##
## Chain 1 finished in 0.1 seconds.
## Chain 2 finished in 0.1 seconds.
## Chain 3 finished in 0.1 seconds.
##
## All 3 chains finished successfully.
## Mean chain execution time: 0.1 seconds.
## Total execution time: 0.5 seconds.Plotting prior and posterior
peake.brm3 %>% get_variables()## [1] "b_Intercept" "b_logAREA" "prior_Intercept" "prior_b"
## [5] "lprior" "lp__" "accept_stat__" "treedepth__"
## [9] "stepsize__" "divergent__" "n_leapfrog__" "energy__"peake.brm3 %>%
hypothesis("logAREA=0") %>%
plot()
peake.brm3 %>% SUYR_prior_and_posterior()
Exploring the stan code
standata(peake.brm3)## $N
## [1] 25
##
## $Y
## [1] 18 60 57 100 48 118 148 214 225 283 380 278 338 274 569
## [16] 509 682 600 978 363 1402 675 1159 1062 632
##
## $K
## [1] 2
##
## $X
## Intercept logAREA
## 1 1 6.246107
## 2 1 6.150731
## 3 1 6.136106
## 4 1 6.844389
## 5 1 7.213142
## 6 1 7.480800
## 7 1 7.430120
## 8 1 7.487896
## 9 1 8.035949
## 10 1 8.289067
## 11 1 8.394989
## 12 1 8.401037
## 13 1 8.513765
## 14 1 8.400853
## 15 1 8.610818
## 16 1 8.919481
## 17 1 8.873303
## 18 1 9.121503
## 19 1 9.223560
## 20 1 9.136445
## 21 1 9.527275
## 22 1 9.918414
## 23 1 10.115073
## 24 1 10.208912
## 25 1 10.170373
## attr(,"assign")
## [1] 0 1
##
## $prior_only
## [1] 0
##
## attr(,"class")
## [1] "standata" "list"stancode(peake.brm3)## // generated with brms 2.18.0
## functions {
##
## }
## data {
## int<lower=1> N; // total number of observations
## array[N] int Y; // response variable
## int<lower=1> K; // number of population-level effects
## matrix[N, K] X; // population-level design matrix
## int prior_only; // should the likelihood be ignored?
## }
## transformed data {
## int Kc = K - 1;
## matrix[N, Kc] Xc; // centered version of X without an intercept
## vector[Kc] means_X; // column means of X before centering
## for (i in 2 : K) {
## means_X[i - 1] = mean(X[ : , i]);
## Xc[ : , i - 1] = X[ : , i] - means_X[i - 1];
## }
## }
## parameters {
## vector[Kc] b; // population-level effects
## real Intercept; // temporary intercept for centered predictors
## }
## transformed parameters {
## real lprior = 0; // prior contributions to the log posterior
## lprior += normal_lpdf(b | 0, 1);
## lprior += normal_lpdf(Intercept | 6, 1.5);
## }
## model {
## // likelihood including constants
## if (!prior_only) {
## target += poisson_log_glm_lpmf(Y | Xc, Intercept, b);
## }
## // priors including constants
## target += lprior;
## }
## generated quantities {
## // actual population-level intercept
## real b_Intercept = Intercept - dot_product(means_X, b);
## // additionally sample draws from priors
## real prior_b = normal_rng(0, 1);
## real prior_Intercept = normal_rng(6, 1.5);
## }MCMC sampling diagnostics
In addition to the regular model diagnostics checking, for Bayesian analyses, it is also necessary to explore the MCMC sampling diagnostics to be sure that the chains are well mixed and have converged on a stable posterior.
There are a wide variety of tests that range from the big picture, overal chain characteristics to the very specific detailed tests that allow the experienced modeller to drill down to the very fine details of the chain behaviour. Furthermore, there are a multitude of packages and approaches for exploring these diagnostics.
rstanarm
bayesplot
The bayesplot package offers a range of MCMC diagnostics
as well as Posterior Probability Checks (PPC), all of which have a
convenient plot() interface. Lets start with the MCMC
diagnostics.
See list of available diagnostics by name
available_mcmc()## bayesplot MCMC module:
## mcmc_acf
## mcmc_acf_bar
## mcmc_areas
## mcmc_areas_ridges
## mcmc_combo
## mcmc_dens
## mcmc_dens_chains
## mcmc_dens_overlay
## mcmc_hex
## mcmc_hist
## mcmc_hist_by_chain
## mcmc_intervals
## mcmc_neff
## mcmc_neff_hist
## mcmc_nuts_acceptance
## mcmc_nuts_divergence
## mcmc_nuts_energy
## mcmc_nuts_stepsize
## mcmc_nuts_treedepth
## mcmc_pairs
## mcmc_parcoord
## mcmc_rank_hist
## mcmc_rank_overlay
## mcmc_recover_hist
## mcmc_recover_intervals
## mcmc_recover_scatter
## mcmc_rhat
## mcmc_rhat_hist
## mcmc_scatter
## mcmc_trace
## mcmc_trace_highlight
## mcmc_violinOf these, we will focus on:
- mcmc_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
plot(peake.rstanarm3, plotfun='mcmc_trace')
The chains appear well mixed and very similar
- acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
plot(peake.rstanarm3, 'acf_bar')
There is no evidence of autocorrelation in the MCMC samples
- Rhat: Rhat is a measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
plot(peake.rstanarm3, 'rhat_hist')
All Rhat values are below 1.05, suggesting the chains have converged.
neff (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
plot(peake.rstanarm3, 'neff_hist')
Ratios all very high.
More diagnostics
plot(peake.rstanarm3, 'combo')
plot(peake.rstanarm3, 'violin')
stan plots
The rstan package offers a range of MCMC diagnostics.
Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
stan_trace(peake.rstanarm3)
The chains appear well mixed and very similar
- stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
stan_ac(peake.rstanarm3) 
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
stan_rhat(peake.rstanarm3) 
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
stan_ess(peake.rstanarm3)
Ratios all very high.
stan_dens(peake.rstanarm3, separate_chains = TRUE)
ggmcmc
The ggmean package also has a set of MCMC diagnostic
functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
peake.ggs <- ggs(peake.rstanarm3)
ggs_traceplot(peake.ggs)
The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
ggs_autocorrelation(peake.ggs)
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
ggs_Rhat(peake.ggs)
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
ggs_effective(peake.ggs)
Ratios all very high.
More diagnostics
ggs_crosscorrelation(peake.ggs)
ggs_grb(peake.ggs)
brms
bayesplot
The bayesplot package offers a range of MCMC diagnostics
as well as Posterior Probability Checks (PPC), all of which have a
convenient plot() interface. Lets start with the MCMC
diagnostics.
See list of available diagnostics by name
available_mcmc()## bayesplot MCMC module:
## mcmc_acf
## mcmc_acf_bar
## mcmc_areas
## mcmc_areas_ridges
## mcmc_combo
## mcmc_dens
## mcmc_dens_chains
## mcmc_dens_overlay
## mcmc_hex
## mcmc_hist
## mcmc_hist_by_chain
## mcmc_intervals
## mcmc_neff
## mcmc_neff_hist
## mcmc_nuts_acceptance
## mcmc_nuts_divergence
## mcmc_nuts_energy
## mcmc_nuts_stepsize
## mcmc_nuts_treedepth
## mcmc_pairs
## mcmc_parcoord
## mcmc_rank_hist
## mcmc_rank_overlay
## mcmc_recover_hist
## mcmc_recover_intervals
## mcmc_recover_scatter
## mcmc_rhat
## mcmc_rhat_hist
## mcmc_scatter
## mcmc_trace
## mcmc_trace_highlight
## mcmc_violinOf these, we will focus on:
- trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different shade of blue, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
mcmc_plot(peake.brm3, type='trace')
The chains appear well mixed and very similar
- acf_bar (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
mcmc_plot(peake.brm3, type='acf_bar')
There is no evidence of autocorrelation in the MCMC samples
- rhat_hist: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
mcmc_plot(peake.brm3, type='rhat_hist')
All Rhat values are below 1.05, suggesting the chains have converged.
neff_hist (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
mcmc_plot(peake.brm3, type='neff_hist')
Ratios all very high.
More diagnostics
mcmc_plot(peake.brm3, type='combo')
mcmc_plot(peake.brm3, type='violin')
stan plots
The rstan package offers a range of MCMC diagnostics.
Lets start with the MCMC diagnostics.
Of these, we will focus on:
- stan_trace: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
stan_trace(peake.brm3$fit)
The chains appear well mixed and very similar
- stan_acf (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
stan_ac(peake.brm3$fit) 
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
stan_rhat(peake.brm3$fit) 
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
stan_ess(peake.brm3$fit)
Ratios all very high.
stan_dens(peake.brm3$fit, separate_chains = TRUE)
ggmcmc
The ggmean package also has a set of MCMC diagnostic
functions. Lets start with the MCMC diagnostics.
Of these, we will focus on:
- ggs_traceplot: this plots the estimates of each parameter over the post-warmup length of each MCMC chain. Each chain is plotted in a different colour, with each parameter in its own facet. Ideally, each trace should just look like noise without any discernible drift and each of the traces for a specific parameter should look the same (i.e, should not be displaced above or below any other trace for that parameter).
peake.ggs <- ggs(peake.brm3)
ggs_traceplot(peake.ggs)
The chains appear well mixed and very similar
- gss_autocorrelation (autocorrelation function): plots the autocorrelation between successive MCMC sample lags for each parameter and each chain
ggs_autocorrelation(peake.ggs)
There is no evidence of autocorrelation in the MCMC samples
- stan_rhat: Rhat is a scale reduction factor measure of convergence between the chains. The closer the values are to 1, the more the chains have converged. Values greater than 1.05 indicate a lack of convergence. There will be an Rhat value for each parameter estimated.
ggs_Rhat(peake.ggs)
All Rhat values are below 1.05, suggesting the chains have converged.
stan_ess (number of effective samples): the ratio of the number of effective samples (those not rejected by the sampler) to the number of samples provides an indication of the effectiveness (and efficiency) of the MCMC sampler. Ratios that are less than 0.5 for a parameter suggest that the sampler spent considerable time in difficult areas of the sampling domain and rejected more than half of the samples (replacing them with the previous effective sample).
If the ratios are low, tightening the priors may help.
ggs_effective(peake.ggs)
Ratios all very high.
More diagnostics
ggs_crosscorrelation(peake.ggs)
ggs_grb(peake.ggs)
Model validation
rstanarm
pp check
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
available_ppc()## bayesplot PPC module:
## ppc_bars
## ppc_bars_grouped
## ppc_boxplot
## ppc_dens
## ppc_dens_overlay
## ppc_dens_overlay_grouped
## ppc_ecdf_overlay
## ppc_ecdf_overlay_grouped
## ppc_error_binned
## ppc_error_hist
## ppc_error_hist_grouped
## ppc_error_scatter
## ppc_error_scatter_avg
## ppc_error_scatter_avg_grouped
## ppc_error_scatter_avg_vs_x
## ppc_freqpoly
## ppc_freqpoly_grouped
## ppc_hist
## ppc_intervals
## ppc_intervals_grouped
## ppc_km_overlay
## ppc_km_overlay_grouped
## ppc_loo_intervals
## ppc_loo_pit
## ppc_loo_pit_overlay
## ppc_loo_pit_qq
## ppc_loo_ribbon
## ppc_ribbon
## ppc_ribbon_grouped
## ppc_rootogram
## ppc_scatter
## ppc_scatter_avg
## ppc_scatter_avg_grouped
## ppc_stat
## ppc_stat_2d
## ppc_stat_freqpoly
## ppc_stat_freqpoly_grouped
## ppc_stat_grouped
## ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed ontop of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
pp_check(peake.rstanarm3, plotfun='dens_overlay')
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
pp_check(peake.rstanarm3, plotfun='error_scatter_avg')
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.
pp_check(peake.rstanarm3, x=peake$AREA, plotfun='error_scatter_avg_vs_x')
- intervals: plots the observed data overlayed ontop of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
pp_check(peake.rstanarm3, x=peake$AREA, plotfun='intervals')
The modelled preditions seem to underestimate the uncertainty with increasing mussel clump area.
- ribbon: this is just an alternative way of expressing the above plot.
pp_check(peake.rstanarm3, x=peake$AREA, plotfun='ribbon')
The shinystan package allows the full suite of MCMC
diagnostics and posterior predictive checks to be accessed via a web
interface.
#library(shinystan)
#launch_shinystan(peake.rstanarm3)DHARMa residuals
DHARMa residuals provide very useful diagnostics. Unfortunately, we
cannot directly use the simulateResiduals() function to
generate the simulated residuals. However, if we are willing to
calculate some of the components ourself, we can still obtain the
simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- posterior_predict(peake.rstanarm3, ndraws=250, summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = peake$INDIV,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE)
plot(peake.resids)
peake.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## dispersion = 95.392, p-value < 2.2e-16
## alternative hypothesis: two.sidedConclusions:
- the simulated residuals suggest a general lack of fit due to overdispersion and outliers
- perhaps we should explore a negative binomial model
brms
pp check
Post predictive checks provide additional diagnostics about the fit of the model. Specifically, they provide a comparison between predictions drawn from the model and the observed data used to train the model.
See list of available diagnostics by name
available_ppc()## bayesplot PPC module:
## ppc_bars
## ppc_bars_grouped
## ppc_boxplot
## ppc_dens
## ppc_dens_overlay
## ppc_dens_overlay_grouped
## ppc_ecdf_overlay
## ppc_ecdf_overlay_grouped
## ppc_error_binned
## ppc_error_hist
## ppc_error_hist_grouped
## ppc_error_scatter
## ppc_error_scatter_avg
## ppc_error_scatter_avg_grouped
## ppc_error_scatter_avg_vs_x
## ppc_freqpoly
## ppc_freqpoly_grouped
## ppc_hist
## ppc_intervals
## ppc_intervals_grouped
## ppc_km_overlay
## ppc_km_overlay_grouped
## ppc_loo_intervals
## ppc_loo_pit
## ppc_loo_pit_overlay
## ppc_loo_pit_qq
## ppc_loo_ribbon
## ppc_ribbon
## ppc_ribbon_grouped
## ppc_rootogram
## ppc_scatter
## ppc_scatter_avg
## ppc_scatter_avg_grouped
## ppc_stat
## ppc_stat_2d
## ppc_stat_freqpoly
## ppc_stat_freqpoly_grouped
## ppc_stat_grouped
## ppc_violin_grouped- dens_overlay: plots the density distribution of the observed data (black line) overlayed ontop of 50 density distributions generated from draws from the model (light blue). Ideally, the 50 realisations should be roughly consistent with the observed data.
pp_check(peake.brm3, type='dens_overlay', ndraws = 100)
The model draws appear deviate from the observed data.
- error_scatter_avg: this plots the observed values against the average residuals. Similar to a residual plot, we do not want to see any patterns in this plot. There is some pattern remaining in these residuals.
pp_check(peake.brm3, type='error_scatter_avg')
The predictive error seems to be related to the predictor - the model performs poorest at higher mussel clump areas.
- error_scatter_avg_vs_x: this is similar to a regular residual plot and as such should be interpreted as such. Again, this is not interpretable for binary data.
pp_check(peake.brm3, x='AREA', type='error_scatter_avg_vs_x')
- intervals: plots the observed data overlayed ontop of posterior predictions associated with each level of the predictor. Ideally, the observed data should all fall within the predictive intervals.
pp_check(peake.brm3, x='AREA', type='intervals')
The modelled preditions seem to underestimate the uncertainty with increasing mussel clump area.
- ribbon: this is just an alternative way of expressing the above plot.
pp_check(peake.brm3, x='AREA', type='ribbon')
The shinystan package allows the full suite of MCMC
diagnostics and posterior predictive checks to be accessed via a web
interface.
#library(shinystan)
#launch_shinystan(peake.brm3)DHARMa residuals
DHARMa residuals provide very useful diagnostics. Unfortunately, we
cannot directly use the simulateResiduals() function to
generate the simulated residuals. However, if we are willing to
calculate some of the components ourself, we can still obtain the
simulated residuals from the fitted stan model.
We need to supply:
- simulated (predicted) responses associated with each observation.
- observed values
- fitted (predicted) responses (averaged) associated with each observation
preds <- posterior_predict(peake.brm3, ndraws=250, summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = peake$INDIV,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE)
plot(peake.resids)
peake.resids %>% testDispersion()
##
## DHARMa nonparametric dispersion test via sd of residuals fitted vs.
## simulated
##
## data: simulationOutput
## dispersion = 96.804, p-value < 2.2e-16
## alternative hypothesis: two.sidedConclusions:
- the simulated residuals suggest a general lack of fit due to overdispersion and outliers
- perhaps we should explore a negative binomial model
Fit Negative Binomial model
rstanarm
Fit the model
peake.rstanarm4 <- stan_glm(INDIV ~ log(AREA), data=peake,
family=neg_binomial_2(),
prior_intercept = normal(6, 2.8, autoscale=FALSE),
prior = normal(0, 2.3, autoscale=FALSE),
prior_aux = rstanarm::exponential(rate=1, autoscale=FALSE),
iter = 5000, warmup = 1000,
chains = 3, thin = 5, refresh = 0
)Plotting prior and posterior
posterior_vs_prior(peake.rstanarm4, color_by='vs', group_by=TRUE,
facet_args=list(scales='free_y'))
Conclusions:
- in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.
## ggemmeans(peake.rstanarm4, ~AREA) %>% plot(add.data=TRUE)
ggpredict(peake.rstanarm4, ~AREA) %>% plot(add.data=TRUE)
MCMC chain and posterior checks
plot(peake.rstanarm4, plotfun='mcmc_trace')
plot(peake.rstanarm4, 'acf_bar')
plot(peake.rstanarm4, 'rhat_hist')
plot(peake.rstanarm4, 'neff_hist')
pp_check(peake.rstanarm4, x=peake$AREA, plotfun='dens_overlay')
pp_check(peake.rstanarm4, x=peake$AREA, plotfun='error_scatter_avg_vs_x')
pp_check(peake.rstanarm4, x=peake$AREA, plotfun='intervals')
DHARMa residuals
preds <- posterior_predict(peake.rstanarm4, ndraws=250, summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = peake$INDIV,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE)
plot(peake.resids)
LOOIC
Different models can be compared via information criterion. LOO (Leave One Out) IC is similar to AIC except that AIC does not consider priors and assumes that the posterior likelihood is multivariate normal. LOO AIC does not and integrates over all uncertainty.
- ELPD: is the theoretical expected log pointwise predictive density for a new dataset and can be estimated via leave-one-out cross-validation.
elpd_loo: is the Bayesian LOO estimate of ELPDSEofelpd_loo: is the Monte Carlo standard error is the estimate for the computational accuracy of MCMC and importance sampling used to computeelpd_loop_loo: is the difference betweenelpd_looand the estimated from non-cross validated ELPD and therefore is a measure of the relative extra difficulty of predicting new data compared to the observed data. It can also be a meaure of the effective number of parameters:- in a well behaved model,
p_loowill be less than the number of estimated parameters and the total sample size. - in a misspecified model,
p_loowill be greater than the number of estimated parameters and the total sample size.
- in a well behaved model,
Pareto k: is a relative measure of the influence of each observation as well as the accuracy with which the associated leave one out calculations can be estimated.- when
k < 0.5: the corresponding components can be accurately estimated - when
0.5 < k < 0.7: the accuracy is lower but still acceptable - when
k>0.7: the accuracy is too low andelpd_loois unreliable. This can also suggest a misspecified model.
- when
More information about the above can be gleaned from [https://mc-stan.org/loo/reference/loo-glossary.html].
Start with the Poisson model
(peake.rstanarm3.loo <- loo(peake.rstanarm3))##
## Computed from 2400 by 25 log-likelihood matrix
##
## Estimate SE
## elpd_loo -933.5 300.1
## p_loo 122.0 48.2
## looic 1867.0 600.2
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 15 60.0% 420
## (0.5, 0.7] (ok) 2 8.0% 128
## (0.7, 1] (bad) 4 16.0% 42
## (1, Inf) (very bad) 4 16.0% 2
## See help('pareto-k-diagnostic') for details.Conclusions:
p_loois substantially higher than the number of estimated parameters- there are some
Pareto kvalues identified as bad or even very bad
Now for the Negative Binomial model
(peake.rstanarm4.loo <- loo(peake.rstanarm4))##
## Computed from 2400 by 25 log-likelihood matrix
##
## Estimate SE
## elpd_loo -156.8 5.3
## p_loo 1.9 0.5
## looic 313.6 10.7
## ------
## Monte Carlo SE of elpd_loo is 0.0.
##
## All Pareto k estimates are good (k < 0.5).
## See help('pareto-k-diagnostic') for details.Conclusions:
p_loois lower than the number of estimated parameters- there are no bad
Pareto kvalues
We can also compare the models.
The difference in Expected Log Pointwise Probability Density
(elpd_dff) computes the difference in elpd_loo
of two models (expressed relative to the higher model). The difference
in elpd_diff will be negative if the expected out-of-sample
predictive accuracy of the first model is higher. If the difference is
be positive then the second model is preferred.
Note, the above assumes that both models are valid. In the current case, we know that the Poisson model was overdispersed and thus it should not be a genuine candidate. Nevertheless, we will compare the Poisson to the Negative Binomial for illustrative purposes.
loo_compare(peake.rstanarm3.loo, peake.rstanarm4.loo)## elpd_diff se_diff
## peake.rstanarm4 0.0 0.0
## peake.rstanarm3 -776.7 296.5Conclusions:
- the
elpd_diffis negative, indicating that the first model (Negative Binomial) is better.
brms
Fit the model
peake.brm4 = brm(bf(INDIV ~ log(AREA), family=negbinomial()),
data=peake,
prior=c(
prior(normal(6, 1.5), class='Intercept'),
prior(normal(0, 1), class='b'),
## prior(gamma(0.01, 0.01), class='shape')
prior(gamma(2, 1), class='shape')
),
sample_prior=TRUE,
iter = 5000, warmup = 1000,
chains = 3, cores = 3,
thin = 5, refresh = 0,
backend = "cmdstan")## Running MCMC with 3 parallel chains...
##
## Chain 1 finished in 0.1 seconds.
## Chain 2 finished in 0.1 seconds.
## Chain 3 finished in 0.1 seconds.
##
## All 3 chains finished successfully.
## Mean chain execution time: 0.1 seconds.
## Total execution time: 0.2 seconds.Plotting prior and posterior
peake.brm4 %>% SUYR_prior_and_posterior()
Conclusions:
- in each case, the prior is substantially wider than the posterior, suggesting that the posterior is not biased towards the prior.
## ggemmeans(peake.brm4, ~AREA) %>% plot(add.data=TRUE)
ggpredict(peake.brm4, ~AREA) %>% plot(add.data=TRUE)
MCMC chain and posterior checks
peake.brm4$fit %>% stan_trace()
peake.brm4$fit %>% stan_ac()
peake.brm4$fit %>% stan_rhat()
peake.brm4$fit %>% stan_ess()
pp_check(peake.brm4, x='AREA',type='dens_overlay', ndraws = 100)
pp_check(peake.brm4, x='AREA', type='intervals')
DHARMa residuals
preds <- posterior_predict(peake.brm4, ndraws=250, summary=FALSE)
peake.resids <- createDHARMa(simulatedResponse = t(preds),
observedResponse = peake$INDIV,
fittedPredictedResponse = apply(preds, 2, median),
integerResponse = TRUE)
plot(peake.resids)
LOOIC
Different models can be compared via information criterion. LOO (Leave One Out) IC is similar to AIC except that AIC does not consider priors and assumes that the posterior likelihood is multivariate normal. LOO AIC does not and integrates over all uncertainty.
- ELPD: is the theoretical expected log pointwise predictive density for a new dataset and can be estimated via leave-one-out cross-validation.
elpd_loo: is the Bayesian LOO estimate of ELPDSEofelpd_loo: is the Monte Carlo standard error is the estimate for the computational accuracy of MCMC and importance sampling used to computeelpd_loop_loo: is the difference betweenelpd_looand the estimated from non-cross validated ELPD and therefore is a measure of the relative extra difficulty of predicting new data compared to the observed data. It can also be a meaure of the effective number of parameters:- in a well behaved model,
p_loowill be less than the number of estimated parameters and the total sample size. - in a misspecified model,
p_loowill be greater than the number of estimated parameters and the total sample size.
- in a well behaved model,
Pareto k: is a relative measure of the influence of each observation as well as the accuracy with which the associated leave one out calculations can be estimated.- when
k < 0.5: the corresponding components can be accurately estimated - when
0.5 < k < 0.7: the accuracy is lower but still acceptable - when
k>0.7: the accuracy is too low andelpd_loois unreliable. This can also suggest a misspecified model.
- when
More information about the above can be gleaned from [https://mc-stan.org/loo/reference/loo-glossary.html].
Start with the Poisson model
(peake.brm3.loo <- loo(peake.brm3))##
## Computed from 2400 by 25 log-likelihood matrix
##
## Estimate SE
## elpd_loo -931.6 299.3
## p_loo 119.9 46.6
## looic 1863.1 598.6
## ------
## Monte Carlo SE of elpd_loo is NA.
##
## Pareto k diagnostic values:
## Count Pct. Min. n_eff
## (-Inf, 0.5] (good) 15 60.0% 477
## (0.5, 0.7] (ok) 2 8.0% 88
## (0.7, 1] (bad) 4 16.0% 26
## (1, Inf) (very bad) 4 16.0% 1
## See help('pareto-k-diagnostic') for details.Conclusions:
p_loois substantially higher than the number of estimated parameters- there are some
Pareto kvalues identified as bad or even very bad
Now for the Negative Binomial model
(peake.brm4.loo <- loo(peake.brm4))##
## Computed from 2400 by 25 log-likelihood matrix
##
## Estimate SE
## elpd_loo -156.4 5.4
## p_loo 2.0 0.5
## looic 312.9 10.9
## ------
## Monte Carlo SE of elpd_loo is 0.0.
##
## All Pareto k estimates are good (k < 0.5).
## See help('pareto-k-diagnostic') for details.Conclusions:
p_loois lower than the number of estimated parameters- there are no bad
Pareto kvalues
We can also compare the models.
The difference in Expected Log Pointwise Probability Density
(elpd_dff) computes the difference in elpd_loo
of two models (expressed relative to the higher model). The difference
in elpd_diff will be negative if the expected out-of-sample
predictive accuracy of the first model is higher. If the difference is
be positive then the second model is preferred.
Note, the above assumes that both models are valid. In the current case, we know that the Poisson model was overdispersed and thus it should not be a genuine candidate. Nevertheless, we will compare the Poisson to the Negative Binomial for illustrative purposes.
loo_compare(peake.brm3.loo, peake.brm4.loo)## elpd_diff se_diff
## peake.brm4 0.0 0.0
## peake.brm3 -775.1 295.6Conclusions:
- the
elpd_diffis negative, indicating that the first model (Negative Binomial) is better.
Partial effects plots
rstanarm
ggpredict
peake.rstanarm4 %>% ggpredict() %>% plot(add.data=TRUE)## $AREA
ggemmeans
## Note, does not seem to backtransform...
peake.rstanarm4 %>% ggemmeans(~AREA, type = "fixed", back.transform = TRUE) %>% plot()
fitted_draws
peake.rstanarm4 %>% fitted_draws(newdata=peake) %>%
median_hdci() %>%
ggplot(aes(x=AREA, y=.value)) +
geom_ribbon(aes(ymin=.lower, ymax=.upper), fill='blue', alpha=0.3) +
geom_line() +
geom_point(data=peake, aes(y=INDIV, x=AREA))
brms
conditional_effects
peake.brm4 %>% conditional_effects() %>% plot(points = TRUE) 
peake.brm4 %>% conditional_effects(spaghetti=TRUE,ndraws=200) %>% plot(points = TRUE) + theme_classic()
## NULLce <- peake.brm4 %>% conditional_effects(spaghetti=TRUE,ndraws=200)
plot(ce, points = TRUE, plot = FALSE)[[1]] + theme_classic()
ggpredict
peake.brm4 %>% ggpredict() %>% plot(add.data=TRUE)## $AREA
ggemmeans
peake.brm4 %>% ggemmeans(~AREA) %>% plot(add.data=TRUE)
epred_draws
peake.brm4 %>% epred_draws(newdata=peake) %>%
median_hdci() %>%
ggplot(aes(x=AREA, y=.epred)) +
geom_ribbon(aes(ymin=.lower, ymax=.upper), fill='blue', alpha=0.3) +
geom_line() +
geom_point(data=peake, aes(y=INDIV, x=AREA))
Model investigation
rstanarm
rstanarm captures the MCMC samples from
stan within the returned list. There are numerous ways to
retrieve and summarise these samples. The first three provide convenient
numeric summaries from which you can draw conclusions, the last four
provide ways of obtaining the full posteriors.
summary
The summary() method generates simple summaries (mean,
standard deviation as well as 10, 50 and 90 percentiles).
summary(peake.rstanarm4)##
## Model Info:
## function: stan_glm
## family: neg_binomial_2 [log]
## formula: INDIV ~ log(AREA)
## algorithm: sampling
## sample: 2400 (posterior sample size)
## priors: see help('prior_summary')
## observations: 25
## predictors: 2
##
## Estimates:
## mean sd 10% 50% 90%
## (Intercept) -1.1 0.7 -2.1 -1.1 -0.2
## log(AREA) 0.8 0.1 0.7 0.8 0.9
## reciprocal_dispersion 4.6 1.3 3.0 4.5 6.4
##
## Fit Diagnostics:
## mean sd 10% 50% 90%
## mean_PPD 479.4 88.4 375.1 472.3 597.2
##
## The mean_ppd is the sample average posterior predictive distribution of the outcome variable (for details see help('summary.stanreg')).
##
## MCMC diagnostics
## mcse Rhat n_eff
## (Intercept) 0.0 1.0 2481
## log(AREA) 0.0 1.0 2481
## reciprocal_dispersion 0.0 1.0 2322
## mean_PPD 1.8 1.0 2349
## log-posterior 0.0 1.0 2328
##
## For each parameter, mcse is Monte Carlo standard error, n_eff is a crude measure of effective sample size, and Rhat is the potential scale reduction factor on split chains (at convergence Rhat=1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 19 raw observations.
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.14. This is the mean of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.32. When the mussel clump is 0, we
expect to find 0.32 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) or 0.71 (median) with a standard deviation of 0. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.82 and 0.82 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
summarise_draws (posterior)
peake.rstanarm4$stanfit %>%
summarise_draws(median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail)We can also alter the CI level.
peake.rstanarm4$stanfit %>%
summarise_draws(median,
~HDInterval::hdi(.x, credMass = 0.9),
rhat, length, ess_bulk, ess_tail)Arguably, it would be better to back-transform to the ratio scale
peake.rstanarm4$stanfit %>%
## tidy_draws() %>%
## exp() %>%
summarise_draws(
~ median(exp(.x)),
~HDInterval::hdi(exp(.x)),
rhat, length, ess_bulk, ess_tail)tidyMCMC
tidyMCMC(peake.rstanarm4$stanfit, estimate.method='median', conf.int=TRUE, conf.method='HPDinterval', rhat=TRUE, ess=TRUE)Conclusions:
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.14. This is the median of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.32. When the mussel clump is 0, we
expect to find 0.32 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.09. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.64 and 0.98 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
as_draws_df (posterior)
peake.rstanarm4$stanfit %>% as_draws_df()## summarised
peake.rstanarm4$stanfit %>%
as_draws_df() %>%
exp() %>%
summarise_draws(median,
HDInterval::hdi,
rhat,
ess_bulk, ess_tail)## summarised on fractional scale
peake.rstanarm4$stanfit %>%
as_draws_df() %>%
dplyr::select(matches('Intercept|AREA')) %>%
summarise_draws(median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail)## OR
names <- peake.rstanarm4 %>%
formula() %>%
model.matrix(peake) %>%
colnames()
peake.rstanarm4$stanfit %>%
as_draws_df() %>%
dplyr::select(any_of(names)) %>%
mutate(across(everything(), exp)) %>%
summarise_draws(median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail)gather_draws
Due to the presence of a log transform in the predictor, it is better to use the regex version.
peake.rstanarm4 %>% get_variables()## [1] "(Intercept)" "log(AREA)" "reciprocal_dispersion"
## [4] "accept_stat__" "stepsize__" "treedepth__"
## [7] "n_leapfrog__" "divergent__" "energy__"peake.draw <- peake.rstanarm4 %>% gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE)
peake.drawWe can then summarise this
peake.draw %>% median_hdcipeake.rstanarm4 %>%
gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
facet_wrap(~.variable, scales='free')
We could alternatively express the parameters on the response scale.
peake.rstanarm4 %>%
gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
group_by(.variable) %>%
mutate(.value=exp(.value)) %>%
median_hdciConclusions:
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.14. This is the median of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.32. When the mussel clump is 0, we
expect to find 0.32 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.64. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.98 and 0.95 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.28. This represents a 128% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
bayesplot
peake.rstanarm4 %>% plot(plotfun='mcmc_intervals') 
tidy_draws
This is purely a graphical depiction on the posteriors.
peake.rstanarm4 %>% tidy_draws()spread_draws
peake.rstanarm4 %>% spread_draws(`.Intercept.*|.*AREA.*`, regex=TRUE)## summarised
peake.rstanarm4 %>%
spread_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
summarise_draws("median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk")## summarised on fractional scale
peake.rstanarm4 %>%
spread_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
mutate(across(everything(), exp)) %>%
summarise_draws("median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk")posterior_samples
peake.rstanarm4 %>% posterior_samples() %>% as_tibble()\(R^2\)
Unfortunately, \(R^2\) calculations
for models other than Gaussian and Binomial have not yet been
implemented for rstanarm models yet.
## peake.rstanarm4 %>% bayes_R2() %>% median_hdcibrms
brms captures the MCMC samples from stan
within the returned list. There are numerous ways to retrieve and
summarise these samples. The first three provide convenient numeric
summaries from which you can draw conclusions, the last four provide
ways of obtaining the full posteriors.
summary
The summary() method generates simple summaries (mean,
standard deviation as well as 10, 50 and 90 percentiles).
summary(peake.brm4)## Family: negbinomial
## Links: mu = log; shape = identity
## Formula: INDIV ~ log(AREA)
## Data: peake (Number of observations: 25)
## Draws: 3 chains, each with iter = 5000; warmup = 1000; thin = 5;
## total post-warmup draws = 2400
##
## Population-Level Effects:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Intercept -1.12 0.70 -2.47 0.29 1.00 2368 1994
## logAREA 0.82 0.08 0.66 0.98 1.00 2390 2306
##
## Family Specific Parameters:
## Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## shape 5.03 1.36 2.85 7.97 1.00 2396 2369
##
## Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).Conclusions:
- in the Model Info, we are informed that the total MCMC posterior sample size is 2400 and that there were 19 raw observations.
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.12. This is the mean of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.33. When the mussel clump is 0, we
expect to find 0.33 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) or 0.98 (median) with a standard deviation of 0.08. The 90% credibility intervals indicate that we are 90% confident that the slope is between 0.82 and 1 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.27. This represents a 127% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
summarise_draws (posterior)
peake.brm4 %>%
summarise_draws(median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail)We can also alter the CI level.
peake.brm4 %>%
summarise_draws(median,
~HDInterval::hdi(.x, credMass = 0.9),
rhat, length, ess_bulk, ess_tail)To narrow down to just the parameters of interest, see the code under the tidy_draws tab.
tidyMCMC
tidyMCMC(peake.brm4$fit, estimate.method='median', conf.int=TRUE, conf.method='HPDinterval', rhat=TRUE, ess=TRUE)Conclusions:
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.13. This is the median of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.32. When the mussel clump is 0, we
expect to find 0.32 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.08. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.66 and 0.98 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.27. This represents a 127% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
as_draws_df (posterior)
peake.brm4 %>% as_draws_df()## summarised
peake.brm4 %>%
as_draws_df() %>%
summarise_draws(
median,
~ HDInterval::hdi(.x),
length,
rhat,
ess_bulk, ess_tail
)## summarised on fractional scale
peake.brm4 %>%
as_draws_df() %>%
dplyr::select(starts_with("b_")) %>%
## mutate(across(everything(), exp)) %>%
exp() %>%
summarise_draws(median,
HDInterval::hdi,
rhat,
length,
ess_bulk, ess_tail)tidy_draws
Return the draws (samples) for each parameter in wide format
peake.brm4 %>% tidy_draws()peake.brm4 %>% get_variables()## [1] "b_Intercept" "b_logAREA" "shape" "prior_Intercept"
## [5] "prior_b" "prior_shape" "lprior" "lp__"
## [9] "accept_stat__" "treedepth__" "stepsize__" "divergent__"
## [13] "n_leapfrog__" "energy__"peake.brm4$fit %>%
tidy_draws() %>%
dplyr::select(matches('^b_.*')) %>%
exp() %>%
summarise_draws(median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail)The above can be useful for exploring the full posteriors of all the parameters of performing specific calculations using the posteriors, summarising the parameters takes a few more steps.
peake.brm4 %>%
tidy_draws() %>%
exp() %>%
gather_variables() %>%
median_hdci()The gather_draws on the other hand, conveniently
combines tidy_draws and gather_variables
together in a single command. Furthermore, it returns all of the
variables. The spread_draws function allows users to target
specific variables (either by naming them in full or via regexp).
gather_draws
Due to the presence of a log transform in the predictor, it is better to use the regex version.
peake.brm4 %>% get_variables()## [1] "b_Intercept" "b_logAREA" "shape" "prior_Intercept"
## [5] "prior_b" "prior_shape" "lprior" "lp__"
## [9] "accept_stat__" "treedepth__" "stepsize__" "divergent__"
## [13] "n_leapfrog__" "energy__"peake.draw <- peake.brm4 %>% gather_draws(`b_.*`, regex=TRUE)
peake.drawWe can then summarise this
peake.draw %>% median_hdcipeake.brm4 %>%
gather_draws(`b_.*`, regex=TRUE) %>%
mutate(.value = exp(.value)) %>%
summarise_draws(median,
HDInterval::hdi,
rhat, length, ess_bulk, ess_tail)peake.brm4 %>%
gather_draws(`b_.*`, regex=TRUE) %>%
ggplot() +
stat_halfeye(aes(x=.value, y=.variable)) +
facet_wrap(~.variable, scales='free')
We could alternatively express the parameters on the response scale.
peake.brm4 %>%
gather_draws(`.Intercept.*|.*AREA.*`, regex=TRUE) %>%
group_by(.variable) %>%
mutate(.value=exp(.value)) %>%
median_hdciConclusions:
- the estimated intercept (expected number of individuals for a mussel
clump of log-transformed area = 0) is -1.13. This is the median of the
posterior distribution for this parameter. If we back-transform this to
the response scale, this becomes 0.32. When the mussel clump is 0, we
expect to find 0.32 individuals in the mussel clump (which obviously
does not exist if it has a log-transformed area of 0!).
- the estimated slope (rate at which the number of individuals changes per 1 unit change in log-transformed area), is 0.82 (mean) with a standard error of 0.66. The 95% credibility intervals indicate that we are 90% confident that the slope is between 0.98 and 0.95 - e.g. there is a significant positive trend. When back-transformed onto the reponse scale, we see that for a one unit increase in log-transformed mussel clump area, the number of individuals increases by a factor of 2.27. This represents a 127% increase per 1 unit change in log-transformed mussel clump area.
- Rhat and number of effective samples for each parameter are also provided as MCMC diagnostics and all look good.
bayesplot
peake.brm4 %>% plot(plotfun='mcmc_intervals') 
spread_draws
peake.brm4 %>% spread_draws(`b_.*`, regex=TRUE)## summarised
peake.brm4 %>%
as_draws_df() %>%
dplyr::select(starts_with("b_")) %>%
summarise_draws("median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk")## summarised on fractional scale
peake.brm4 %>%
as_draws_df() %>%
dplyr::select(starts_with("b_")) %>%
mutate(across(everything(), exp)) %>%
summarise_draws("median",
~ HDInterval::hdi(.x),
"rhat",
"ess_bulk")posterior_samples
peake.brm4 %>% posterior_samples() %>% as_tibble()\(R^2\)
peake.brm4 %>% bayes_R2(summary=FALSE) %>% median_hdciHypothesis testing
Since we have the entire posterior, we are able to make probablity statements. We simply count up the number of MCMC sample draws that satisfy a condition (e.g represent a slope greater than 0) and then divide by the total number of MCMC samples.
For this exersize, we will explore the following:
- what this the probability that the number of individuals increases with increasing mussel clump area - this is just asking what proportion of the drawn slopes are greater than 0.
- how much of a change in individuals do we expect if the mussel clump area is increased from 5,000 to 10,000 units and what is the probability that the change is more than 50%
rstanarm
Prob. of effect
Unfortunately, the inline log-transformation of the predictor
interfers with hypothesis()’s ability to find the slope. We
can get around this by renaming before calling
hypothesis().
peake.rstanarm4 %>% as.data.frame() %>% rename(lAREA=`log(AREA)`) %>% hypothesis('lAREA>0')## Hypothesis Tests for class :
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
## 1 (lAREA) > 0 0.82 0.09 0.68 0.97 Inf 1 *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.Conclusions:
- the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
- the evidence ratio is normally the ratio of the number of cases that satisty the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)
Alternatively…
peake.rstanarm4 %>% tidy_draws() %>% summarise(P=sum(`log(AREA)`>0)/n())## # A tibble: 1 × 1
## P
## <dbl>
## 1 1Conclusions:
- the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
Change 5000 to 10000 (Method 1)
newdata = list(AREA=c(5000, 10000))
## fractional scale
peake.rstanarm4 %>% emmeans(~AREA, at=newdata, type = 'response') %>% pairs(reverse = TRUE)## contrast ratio lower.HPD upper.HPD
## AREA10000 / AREA5000 1.77 1.56 1.98
##
## Point estimate displayed: median
## Results are back-transformed from the log scale
## HPD interval probability: 0.95## absolute scale
peake.rstanarm4 %>% emmeans(~AREA, at=newdata) %>% regrid() %>% pairs(reverse = TRUE)## contrast estimate lower.HPD upper.HPD
## AREA10000 - AREA5000 271 183 379
##
## Point estimate displayed: median
## HPD interval probability: 0.95Conclusions:
- the change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 0.57.
If we want to derive other properties, such as the percentage change,
then we use tidy_draws() and then simple
tidyverse spreadsheet operation.
peake.mcmc <- peake.rstanarm4 %>% emmeans(~AREA, at=newdata) %>%
regrid() %>%
tidy_draws() %>%
rename_with(~str_replace(., 'AREA ', 'p')) %>%
mutate(Eff=p10000 - p5000,
PEff=100*Eff/p5000)
peake.mcmc %>% headNow we can calculate the medians and HPD intervals of each column
(and ignore the .chain, .iteration and
.draw).
peake.mcmc %>% tidyMCMC(estimate.method='median',
conf.int=TRUE, conf.method='HPDinterval')Alternatively, we could use median_hdci
peake.mcmc %>% median_hdci(PEff)Conclusions:
- the percentage change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 77.08% with a 95% credibility interval of 55.65 97.6
To get the probability that the effect is greater than a 50% increase.
peake.mcmc %>% summarise(P=sum(PEff>50)/n())## # A tibble: 1 × 1
## P
## <dbl>
## 1 0.996Conclusions:
- the probability that the number of individuals will increase by more
than 50% following an increase in mussel clump area from 5,000 to 10,000
is
Finally, we could alternatively use hypothesis(). Note
that when we do so, the estimate is the difference between the effect
and the hypothesised effect (50%).
peake.mcmc %>% hypothesis('PEff>50')## Hypothesis Tests for class :
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0 27.23 10.6 10.52 45.28 239 1
## Star
## 1 *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.Change 5000 to 10000 (Method 2)
## Note, the P value is on absolute difference
newdata = list(AREA=c(5000, 10000))
peake.rstanarm4 %>%
emmeans(~AREA, at=newdata) %>%
regrid() %>%
pairs(reverse = TRUE) %>%
tidy_draws() %>%
summarise(across(contains('contrast'),
list(P = ~ sum(.>50)/n(),
HDCI = ~ median_hdci(.)),
.names = c('{.fn}')
)) %>%
tidyr::unpack(HDCI)We can also express this as a percentage change
newdata = list(AREA=c(5000, 10000))
## Simple
peake.rstanarm4 %>%
emmeans(~AREA, at=newdata) %>%
pairs(reverse = TRUE) %>%
regrid() ## contrast ratio lower.HPD upper.HPD
## AREA10000/AREA5000 1.77 1.56 1.98
##
## Point estimate displayed: median
## HPD interval probability: 0.95## More advanced (both P and percent change)
peake.mcmc <- peake.rstanarm4 %>%
emmeans(~AREA, at=newdata) %>%
pairs(reverse = TRUE) %>%
regrid() %>%
tidy_draws() %>%
mutate(across(contains('contrast'), ~ 100*(. - 1)))
peake.mcmc %>%
summarise(across(contains('contrast'),
list(P = ~ sum(.>50)/n(),
HDCI = ~ median_hdci(.)),
.names = c('{.fn}')
)) %>%
tidyr::unpack(HDCI)Change 5000 to 10000 (Method 3)
peake.rstanarm4 %>%
linpred_draws(newdata = as.data.frame(newdata)) %>%
mutate(.linpred = exp(.linpred)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.linpred),
PEff = 100*Eff/.linpred[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()##OR
peake.rstanarm4 %>%
epred_draws(newdata = as.data.frame(newdata)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.epred),
PEff = 100*Eff/.epred[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()##OR for prediction of individual values
peake.rstanarm4 %>%
predicted_draws(newdata = as.data.frame(newdata)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.prediction),
PEff = 100*Eff/.prediction[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()brm
Prob. of effect
peake.brm4 %>% hypothesis('logAREA>0')## Hypothesis Tests for class b:
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob Star
## 1 (logAREA) > 0 0.82 0.08 0.68 0.96 Inf 1 *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.Conclusions:
- the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
- the evidence ratio is normally the ratio of the number of cases that satisty the hypothesis to the number of cases that do not. Since the number of cases that do not satisfy the hypothesis is 0, the evidence ratio is Inf (since division by 0)
Alternatively…
peake.brm4 %>% tidy_draws() %>% summarise(P=sum(b_logAREA>0)/n())## # A tibble: 1 × 1
## P
## <dbl>
## 1 1Conclusions:
- the probability of a positive effect of mussel clump area on number of individuals is 1. That is, there is, we are 100% confident that there is an effect.
Change 5000 to 10000 (Method 1)
newdata = list(AREA=c(5000, 10000))
## fractional scale
peake.brm4 %>% emmeans(~AREA, at=newdata, type = 'response') %>% pairs(reverse = TRUE)## contrast ratio lower.HPD upper.HPD
## AREA10000 / AREA5000 1.77 1.58 1.97
##
## Point estimate displayed: median
## Results are back-transformed from the log scale
## HPD interval probability: 0.95## absolute scale
peake.brm4 %>% emmeans(~AREA, at=newdata) %>% regrid() %>% pairs(reverse = TRUE)## contrast estimate lower.HPD upper.HPD
## AREA10000 - AREA5000 269 179 369
##
## Point estimate displayed: median
## HPD interval probability: 0.95Conclusions:
- the change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 0.57.
If we want to derive other properties, such as the percentage change,
then we use tidy_draws() and then simple
tidyverse spreadsheet operation.
peake.mcmc <- peake.brm4 %>% emmeans(~AREA, at=newdata) %>%
regrid() %>%
tidy_draws() %>%
rename_with(~str_replace(., 'AREA ', 'p')) %>%
mutate(Eff=p10000 - p5000,
PEff=100*Eff/p5000)
peake.mcmc %>% headNow we can calculate the medians and HPD intervals of each column
(and ignore the .chain, .iteration and
.draw).
peake.mcmc %>% tidyMCMC(estimate.method='median',
conf.int=TRUE, conf.method='HPDinterval')Alternatively, we could use median_hdci
peake.mcmc %>% median_hdci(PEff)Conclusions:
- the percentage change in expected number of individuals associated with an increase in mussel clump area from 5,000 to 10,000 is 76.51% with a 95% credibility interval of 57.56 97.48
To get the probability that the effect is greater than a 50% increase.
peake.mcmc %>% summarise(P=sum(PEff>50)/n())## # A tibble: 1 × 1
## P
## <dbl>
## 1 0.995Conclusions:
- the probability that the number of individuals will increase by more
than 50% following an increase in mussel clump area from 5,000 to 10,000
is
Finally, we could alternatively use hypothesis(). Note
that when we do so, the estimate is the difference between the effect
and the hypothesised effect (50%).
peake.mcmc %>% hypothesis('PEff>50')## Hypothesis Tests for class :
## Hypothesis Estimate Est.Error CI.Lower CI.Upper Evid.Ratio Post.Prob
## 1 (PEff)-(50) > 0 26.78 10.17 10.59 44.01 217.18 1
## Star
## 1 *
## ---
## 'CI': 90%-CI for one-sided and 95%-CI for two-sided hypotheses.
## '*': For one-sided hypotheses, the posterior probability exceeds 95%;
## for two-sided hypotheses, the value tested against lies outside the 95%-CI.
## Posterior probabilities of point hypotheses assume equal prior probabilities.Change 5000 to 10000 (Method 2)
## Note, the P value is on absolute difference
newdata = list(AREA=c(5000, 10000))
peake.brm4 %>%
emmeans(~AREA, at=newdata) %>%
regrid() %>%
pairs(reverse = TRUE) %>%
tidy_draws() %>%
summarise(across(contains('contrast'),
list(P = ~ sum(.>50)/n(),
HDCI = ~ median_hdci(.)),
.names = c('{.fn}')
)) %>%
tidyr::unpack(HDCI)We can also express this as a percentage change
newdata = list(AREA=c(5000, 10000))
## Simple
peake.brm4 %>%
emmeans(~AREA, at=newdata) %>%
pairs(reverse = TRUE) %>%
regrid() ## contrast ratio lower.HPD upper.HPD
## AREA10000/AREA5000 1.77 1.58 1.97
##
## Point estimate displayed: median
## HPD interval probability: 0.95## More advanced (both P and percent change)
peake.mcmc <- peake.brm4 %>%
emmeans(~AREA, at=newdata) %>%
pairs(reverse = TRUE) %>%
regrid() %>%
tidy_draws() %>%
mutate(across(contains('contrast'), ~ 100*(. - 1)))
peake.mcmc %>%
summarise(across(contains('contrast'),
list(P = ~ sum(.>50)/n(),
HDCI = ~ median_hdci(.)),
.names = c('{.fn}')
)) %>%
tidyr::unpack(HDCI)Change 5000 to 10000 (Method 3)
peake.brm4 %>%
linpred_draws(newdata = as.data.frame(newdata)) %>%
mutate(.linpred = exp(.linpred)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.linpred),
PEff = 100*Eff/.linpred[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()##OR
peake.brm4 %>%
epred_draws(newdata = as.data.frame(newdata)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.epred),
PEff = 100*Eff/.epred[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()##OR for prediction of individual values
peake.brm4 %>%
predicted_draws(newdata = as.data.frame(newdata)) %>%
ungroup() %>%
group_by(.draw) %>%
summarise(Eff = diff(.prediction),
PEff = 100*Eff/.prediction[1]) %>%
ungroup() %>%
mutate(P = sum(PEff>50)/n()) %>%
pivot_longer(cols = -.draw) %>%
group_by(name) %>%
median_hdci()Summary figure
rstanarm
## Using emmeans
peake.grid = with(peake, list(AREA = seq(min(AREA), max(AREA), len=100)))
newdata = emmeans(peake.rstanarm4, ~AREA, at=peake.grid, type='response') %>% as.data.frame
head(newdata)ggplot(newdata, aes(y=prob, x=AREA)) +
geom_point(data=peake, aes(y=INDIV)) +
geom_line() +
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD), fill='blue', alpha=0.3) +
scale_y_continuous('Individuals') +
scale_x_continuous('Mussel clump area') +
theme_classic()
## spaghetti plot
newdata = emmeans(peake.rstanarm4, ~AREA, at=peake.grid) %>%
regrid() %>%
gather_emmeans_draws()
newdata %>% headggplot(newdata, aes(y=.value, x=AREA)) +
geom_line(aes(group=.draw), colour = 'orange', alpha=0.01) +
geom_point(data=peake, aes(y=INDIV)) +
scale_y_continuous('Number of Individuals') +
scale_x_continuous(expression(Mussel~clump~area~(per~mm^2))) +
theme_classic()
brms
## Using emmeans
peake.grid = with(peake, list(AREA = seq(min(AREA), max(AREA), len=100)))
newdata = emmeans(peake.brm4, ~AREA, at=peake.grid, type='response') %>% as.data.frame
head(newdata)ggplot(newdata, aes(y=prob, x=AREA)) +
geom_point(data=peake, aes(y=INDIV)) +
geom_line() +
geom_ribbon(aes(ymin=lower.HPD, ymax=upper.HPD), fill='blue', alpha=0.3) +
scale_y_continuous('Individuals') +
scale_x_continuous('Mussel clump area') +
theme_classic()
## spaghetti plot
newdata = emmeans(peake.brm4, ~AREA, at=peake.grid) %>%
regrid() %>%
gather_emmeans_draws()
newdata %>% headggplot(newdata, aes(y=.value, x=AREA)) +
geom_line(aes(group=.draw), colour = 'orange', alpha=0.01) +
geom_point(data=peake, aes(y=INDIV)) +
scale_y_continuous('Number of Individuals') +
scale_x_continuous(expression(Mussel~clump~area~(per~mm^2))) +
theme_classic()